Randall Munroe has given a very good summary of the argumentation:
Humans will go extinct someday. Suppose that, after this happens, aliens somehow revive all humans who have ever lived. They line us up in order of birth and number us from 1 to N. Then they divide us divide them into three groups--the first 5%, the middle 90%, and the last 5%:
Now imagine the aliens ask each human (who doesn't know how many people lived after their time), "Which group do you think you're in?"
Most of them probably wouldn't speak English, and those who did would probably have an awful lot of questions of their own. But if for some reason every human answered "I'm in the middle group", 90% of them will (obviously) be right. This is true no matter how big N is.
Therefore, the argument goes, we should assume we're in the middle 90% of humans. Given that there have been a little over 100 billion humans so far, we should be able to assume with 95% probability that N is less than 2.2 trillion humans. If it's not, it means we're assuming we're in 5% of humans--and if all humans made that assumption, most of them would be wrong.
To put it more simply: Out of all people who will ever live, we should probably assume we're somewhere in the middle; after all, most people are.
If our population levels out around 9 billion, this suggests humans will probably go extinct in about 800 years, and not more than 16,000.He goes on to state that most people immediately conclude that the idea is obviously wrong, but "the problem is, everyone thinks it's wrong for a different reason. And the more they study it, the more they tend to change their minds about what that reason is."
Well, there are two reasons why that could be so. One is that the argument is really quite clever but most people don't realise it. The other is that there is so much wrong with it that people discover new layers of wrongness every time they look at it.
I guess I would have to be counted among those who think that the Doomsday Argument is, indeed, idiotic. Admittedly I cannot come up with a super-deep Bayesian counter-argument such as are referenced in the linked Wikipedia article. But I don't think that is necessary because this does not look like a job for probabilistic reasoning anyway.